Three ways to describe the middle
When you have a list of numbers, you often want one value that stands for the lot. There are three common choices. They're called the mean, the median and the mode. Each tells a slightly different story, and picking the wrong one can mislead you.
Paste your numbers in, separated by commas, and the calculator shows all three along with the range, the sum and the count. Each step is written out, so students can see how the answer was built.
Mean: the average everyone knows
Add everything up and divide by how many numbers there are. For 4, 8, 6, 5 and 7 the sum is 30 and there are five values, so the mean is 6.
The mean uses every number, which is its strength and its weakness. One huge value pulls it hard. If four people earn 40,000 and one earns 400,000, the mean income is 112,000. That doesn't describe anyone in the group well.
Median: the middle one
Sort the list and pick the center. With an odd count, it's the middle value. With an even count, average the two middle values. For 3, 5, 9, 12 the median is (5 + 9) / 2, which is 7.
Outliers barely move the median. In the income example above, it stays at 40,000. That's why house prices and salaries are usually reported as medians.
Mode: the most common
The mode is the value that appears most often. In 2, 3, 3, 5, 7, 3 it's 3. A set can have two modes, or more, if values tie. It can also have none, when every number appears once.
Mode is the only one of the three that works for non-numeric data, like the most common shoe size sold or the most picked color. In a number list, it's handy for spotting repeat values.
Range: how spread out is it
Range is the highest value minus the lowest. For 3, 5, 9, 12 it's 9. It's quick but crude, since a single odd value stretches it. Still, it gives a sense of the spread right away.
Two sets can share the same mean and have totally different ranges. Test scores of 50 and 90 average 70, and so do scores of 69 and 71. The range shows the difference.
Which one should you use?
If the data is fairly even, the mean is fine. If there are extreme values, lean on the median. If you care about what's most typical or popular, use the mode. When in doubt, look at all three. Big gaps between them are a hint that the data is lopsided.
If your mean doesn't match what you expected, check for a typo, a stray extra comma or a value entered twice. The step view helps catch that.
Your numbers never leave your browser, so it's safe to paste real grades or sales data.
A small worked example
Take seven salaries in a small office: 32, 34, 35, 36, 38, 40 and 250, all in thousands. The mean is 66.4. That's higher than six of the seven people earn, because one boss drags it up. The median is 36, the fourth value in order, and it describes the typical worker far better.
This is why reports about income and house prices lean on the median. A handful of huge values can bend the mean badly while the median barely moves.
No value repeats here, so there's no mode. That's allowed. A set can have one mode, several or none.
Ties, even counts and odd cases
With an even number of values, the median is the average of the two middle ones. For 4, 8, 10, 20 it's 9. Some people expect a value from the list, but 9 isn't in it, and that's fine.
Two values sharing the top count make the set bimodal. Shoe sizes in a shop often do that, one peak for women's sizes and another for men's.
Always sort before picking the middle. Skipping that step is the most common error on homework.
Where each one earns its place
Use the mean for symmetric data without wild outliers, like test scores across a big class. Use the median for skewed data such as wages, rents and delivery times. Use the mode for categories, like the most ordered item on a menu.
The range, which is the largest minus the smallest, gives a quick feel for spread. It's sensitive to outliers too, so read it next to the median.
Paste your numbers, check the working and compare all four before you decide what the data says.